Chapter 47 introduces the components that are needed to specify a dynamical system and the formats by which the action of the system is displayed.
The components are:
The changing quantities, functions of time whose instantaneous value describes the instantaneous configuration of the system. The set of these quantities at a given instant constitutes the state (or dynamical state) of the system. For a pendulum, for instance, the state has two components: the instantaneous angle that the bob makes with the vertical and the time rate of change of that angle.
The state space, each point of which is a possible instantaneous state.
The dynamical rule, which assigns to each and every point in state space a vector. You can think of the state as a kind of generalized “position” and the dynamical rule as the assignment of a “velocity” vector that can vary from place to place in state space.
It’s crucial that students understand the distinction between state, state spce, dynamical rule, trajectory, and time series. So take time to have students practice with the ideas. It’s likely you’ll need two class days for this chapter.
Let’s present these components using a children’s game, Chutes and Ladders, as the illustration.
The game board is the set of possible positions, that is, the state space. In chutes and ladders, there are 100 numbered squares on the board so there are only 100 possible values of the state. (For our study of dynamics, the state space will typically be a continuous coordinate plane, on which you can specify position by the horizontal and vertical coordinates.)
The player has a token that marks the instantaneous state for that player.
The underlying dynamical rule is that the player rolls a die and moves the corresponding number of squares forward. But the dynamical systems we will study are deterministic, there is no randomness involved (until Chapter 55). So to make the analogy work, let’s replace the roll-and-move logic with something much simpler: Each round will consist of moving the state from its current square to the next numbered square.
But the dynamical rule is somewhat more complicated than “move one square forward.” From place to place there are ladders that connect two squares. When the state reaches a square holding the foot of a ladder, the state is swept up to the higher-numbered square at the top of the ladder. Similarly, there are chutes. These work much like the ladders but carry the state from a higher-numbered square to a lower-numbered square.
The small drawings on the board are not actually part of the action of the game. Rather, they represent the idea that good deeds lead the player to progress, while wrong-doing produces regression. Thus, the productive gardener in square 1 is rewarded by being moved upward to the harvest in square 38. In square 64 a brat is pulling on his sister’s braids. This misdeed results in punishment: he is moved back to square 60.
A good class activity is to calculate the movement of one player, starting at square 2. The movement is simple, but carry it forward for a dozen or more steps.
\(2 \implies 3 \implies 14 \implies 15 \implies 16 \implies 6 \implies 7 \implies 8 \implies 31 \implies 32 \implies 33 \implies 34 \implies 35 \implies 44 \implies \cdots\)
The starting position (square 2 in this example) is called the initial condition. It might better be called the “intial state,” but we’re sticking to the language conventions used in dynamics. The sequence of states is the trajectory followed from the initial condition.
As described, this is a very simple-minded game. The dynamical systems we’re going to study are similarly simple-minded: just follow the rules.
Still, there are consequences of the chutes-and-ladder dynamics that are not immediately obvious when looking at the board. For instance, is it possible to enter into an unending cycle of movement? Are there some initial conditions that will lead eventually to completion of the game (at square 100), and others that won’t?
Our dynamical systems will have a state space that is:
We will specify the dynamics—the rules of the game—using two equivalent formats:
In the case of (ii), we use a standard framework called a differential equation for reminding us which dynamical function governs which component of the flow. For instance, we might have the system of two differential equations \[\partial_t x = f(x, y)\\ \partial_t y = g(x, y)\]
\(f()\) and \(g()\) are the dynamical functions. Each takes the coordinates of the state as an input. The notation \(\partial_t x =\) is just a way of saying that \(f(x,y)\) is the rule for the \(x\)-component of the flow, while \(\partial_t y =\) makes it clear that \(g(x,y)\) is the \(y\)-component of the flow.
Collectively move one player along the board.
Note: Don’t worry about the equations of motion yet. That’s a topic for another chapter.
Students draw a flow field on a one-dimensional state space: the phase line. Their flow field can be whatever they like, but they should have in mind the flow being a continuous function of position on the line, even if they can only draw a handful of vectors in their diagram.
Znotes::phase_line(x*(1-x) ~ x, domain(x=-0.1:1.1), nix_dyn=FALSE)
6. Translate a trajectory into a graph of \(x\) versus time. This is called a time series plot. The function that is plotted is a solution to the differential equation. Can there be more than one solution to a differential equation? (Ans: Yes, each initial condition has its own solution.)
Draw a square around a region on the board. Keep it blank, with no flow field drawn.
We’re going to be writing differential equations like this: \[\partial_t x = f(x)\]
The derivative on the left side is always with respect to time.
\(x\) really stands for \(x(t)\), a function of time. We drop the \((t)\) only because it makes the equation harder to read: \[\partial_t x(t) = f(x(t))\]
In dynamics we have two different types of functions that play different roles:
The dynamical functions are the rules of the game. The solution functions are a movie showing the action of the game, that is, how the state changes in time.